Proof: ChooseSinceis continuous onis continuous atfor eachThus for eachthere issuch that ifandthenConsider the family of intervalsThis is an open cover ofandis compact so there is a finite subcover ofso there aresuch that

Letand supposeandthen there is such thatNowmaking

Henceandis uniformly continuous.

This theorem has a converse. Supposeis uniformly continuous thenhas a limit at each accumulation point ofLetbe the set of accumulation points ofand let defineDefinebyforandfor The functionis continuous.